Struct FreePatchFlowBasis 

pub struct FreePatchFlowBasis { /* private fields */ }

Truncated polynomial vector-field basis on a d = 2 Euclidean patch, v_{c,(a,b)}(t) = e_c · u₀^a · u₁^b where u = (t − center) ⊙ inv_half ∈ [−1, 1]² is the affinely-normalized patch coordinate (conditioning: the raw t box can be far from [−1, 1]). Because the patch is contractible there is no hairy-ball obstruction — these are smooth global vector fields, so the flow φ_θ(t) = t + Σ_k θ_k v_k(t) parameterizes a neighborhood of the identity in Diff(patch) exactly the way the Fourier basis does on the torus, and any such map with det Dφ > 0 on the patch is a diffeomorphism onto its image. The reference metric is the flat g_ref = I (uniform-speed coordinates), so the canonical chart is the minimum-anisotropy-defect one.

Implementations

impl FreePatchFlowBasis

pub fn new(lo: [f64; 2], hi: [f64; 2]) -> Result<Self, String>

Build the basis for the patch box [lo, hi] per axis. Returns an error when an axis has collapsed (hi ≤ lo) — a patch with no extent in some direction has no honest flat chart.

pub fn dim(&self) -> usize

Number of flow coefficients θ: 2 components × #monomials.

pub fn mode_layout(&self) -> Vec<PatchFlowModeKey>

Mode identities in coefficient order (for each component, each monomial). This IS the θ index layout — Self::mode_samples matches it.

pub fn mode_samples(&self, t: [f64; 2]) -> Vec<FlowModeSample>

Sample every mode (value + gradient in t) at chart point t, in θ order. The monomial f(u) = u₀^a · u₁^b has ∂f/∂t_d = (∂f/∂u_d)·(∂u_d/∂t_d) = (∂f/∂u_d)·inv_half[d] by the chain rule, so the returned gradient is already in the chart coordinate the flow Jacobian Dφ = I + Σ θ_k ∂v_k/∂t lives in.

pub fn map_point(&self, theta: &[f64], t: [f64; 2]) -> [f64; 2]

φ_θ(t) = t + Σ_k θ_k v_k(t) (no wrap — the patch is not periodic).

pub fn flow_jacobian(&self, theta: &[f64], t: [f64; 2]) -> [[f64; 2]; 2]

Flow Jacobian Dφ_θ(t) = I + Σ_k θ_k Dv_k(t), row-major.

pub fn min_jacobian_det_on_grid(&self, theta: &[f64]) -> f64

Minimum of det Dφ_θ over the widened normalized check grid [−1.1, 1.1]² (in t coordinates) — the diffeomorphism guard’s decision quantity. The grid is built in t by inverting the normalization.

Trait Implementations

impl Clone for FreePatchFlowBasis

fn clone(&self) -> FreePatchFlowBasis

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for FreePatchFlowBasis

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.